Symmetries in central-force problems

The two-body problem in central fields (reducible to a central-force problem) models a lot of concrete astronomical situations. The corresponding vector fields (in Cartesian and polar coordinates, extended via collision-blow-up and infinity-blow-up transformations) exhibit nice symmetries that form eight-element Abelian groups endowed with an idempotent structure. All these groups are isomorphic, which is not a trivial result, given the different structures of the corresponding phase spaces. Each of these groups contains seven four-element subgroups isomorphic to Klein?s group. These symmetries are of much help in understanding various characteristics of the global flow of the general problem or of a concrete problem at hand, and are essential in searching for periodic orbits. Problem dva tela za centralne sile (svodljiv na problem centralnog kretanja cestice) daje model za mnoge astronomske situacije. Odgovarajuca vektorska polja (u Dekartovim i polarnim koordinatama, razradjena preko transformacija odstranjivanjem kolizije i beskonacnosti) pokazuju lepe simetrije koje obrazuju osmoclane abelovske grupe sa idempotentnom strukturom. Sve ove grupe su izomorfne sto nije trivijalan rezultat kada su date razlicite strukture odgovarajucih faznih prostora. Svaka od ovih grupa sadrzi sedam podgrupa od po cetiri clana izomorfne sa Klajnovom grupom. Ove simetrije su od velike pomoci u razumevanju raznih karakteristika globalnog dotoka opsteg problema ili konkretnog problema koji se resava i, osim toga, su od sustinskog znacaja u traganju za periodicnim orbitama.